profvis demo

n = 1e6
d = tibble(
  x1 = rt(n, df = 3),
  x2 = rt(n, df = 3),
  x3 = rt(n, df = 3),
  x4 = rt(n, df = 3),
  x5 = rt(n, df = 3),
) |>
  mutate(y = -2*x1 - 1*x2 + 0*x3 + 1*x4 + 2*x5 + rnorm(n))

profvis::profvis(lm(y~., data=d))

Benchmarking - bench

d = tibble(
  x = runif(10000),
  y = runif(10000)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))

# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]           128µs    140µs     7002.  239.84KB     19.4
2 d[which(d$x > 0.5), ]    137µs    152µs     6388.  271.49KB     33.5
3 subset(d, x > 0.5)       167µs    195µs     4940.  288.85KB     26.3
4 filter(d, x > 0.5)       379µs    432µs     2167.    1.48MB     35.4

Larger n

d = tibble(
  x = runif(1e6),
  y = runif(1e6)
)

(b = bench::mark(
  d[d$x > 0.5, ],
  d[which(d$x > 0.5), ],
  subset(d, x > 0.5),
  filter(d, x > 0.5)
))

# A tibble: 4 × 6
  expression                 min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 d[d$x > 0.5, ]            12ms   12.4ms      80.6    13.4MB     60.4
2 d[which(d$x > 0.5), ]   13.3ms   13.6ms      73.6    24.8MB    210. 
3 subset(d, x > 0.5)      17.7ms   17.9ms      55.7    24.8MB     65.0
4 filter(d, x > 0.5)      13.5ms   13.9ms      71.9    24.8MB     56.8

bench - relative results

summary(b, relative=TRUE)

# A tibble: 4 × 6
  expression              min median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>            <dbl>  <dbl>     <dbl>     <dbl>    <dbl>
1 d[d$x > 0.5, ]         1      1         1.45      1        1.06
2 d[which(d$x > 0.5), ]  1.11   1.09      1.32      1.86     3.70
3 subset(d, x > 0.5)     1.47   1.45      1         1.86     1.14
4 filter(d, x > 0.5)     1.12   1.12      1.29      1.86     1   

t.test

Imagine we have run 1000 experiments (rows), each of which collects data on 50 individuals (columns). The first 25 individuals in each experiment are assigned to group 1 and the rest to group 2.

The goal is to calculate the t-statistic for each experiment comparing group 1 to group 2.

m = 1000
n = 50
X = matrix(
  rnorm(m * n, mean = 10, sd = 3), 
  ncol = m
) |>
  as.data.frame() |>
  set_names(paste0("exp", seq_len(m))) |>
  mutate(
    ind = seq_len(n),
    group = rep(1:2, each = n/2)
  ) |>
  as_tibble() |>
  relocate(ind, group)
X

# A tibble: 50 × 1,002
     ind group  exp1  exp2  exp3  exp4  exp5  exp6
   <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
 1     1     1 12.1  15.0  12.8   5.54 12.0  11.5 
 2     2     1  5.61  6.99 14.3   8.47  6.49  5.17
 3     3     1 12.5  11.6  15.7  10.3  14.0   7.52
 4     4     1  8.33  8.39 10.5  15.6   9.94  8.21
 5     5     1 11.1   5.80  8.22  9.48  5.08  2.59
 6     6     1  9.48  6.19  6.05  9.16  8.69 17.0 
 7     7     1 11.2  11.8   5.98  6.05  9.70 12.9 
 8     8     1  9.56  8.14 14.6  11.4   7.39  9.63
 9     9     1 10.1   8.72 10.3   7.71 11.5  12.1 
10    10     1 17.1  11.6   8.38 12.3   7.88  8.59
# ℹ 40 more rows
# ℹ 994 more variables: exp7 <dbl>, exp8 <dbl>,
#   exp9 <dbl>, exp10 <dbl>, exp11 <dbl>,
#   exp12 <dbl>, exp13 <dbl>, exp14 <dbl>,
#   exp15 <dbl>, exp16 <dbl>, exp17 <dbl>,
#   exp18 <dbl>, exp19 <dbl>, exp20 <dbl>,
#   exp21 <dbl>, exp22 <dbl>, exp23 <dbl>, …

Implementations

ttest_formula = function(X, m) {
  for(i in 1:m) t.test(X[[2+i]] ~ X$group)$stat
}

system.time(ttest_formula(X,m))

   user  system elapsed 
  0.183   0.001   0.186 

ttest_for = function(X, m) {
  for(i in 1:m) t.test(X[[2+i]][X$group == 1], X[[2+i]][X$group == 2])$stat
}

system.time(ttest_for(X,m))

   user  system elapsed 
  0.064   0.001   0.066 

ttest_apply = function(X) {
  f = function(x, g) {
    t.test(x[g==1], x[g==2])$stat
  }
  apply(X[,-(1:2)], 2, f, X$group)
}

system.time(ttest_apply(X))

   user  system elapsed 
  0.053   0.000   0.054 

Implementations (cont.)

ttest_hand_calc = function(X) {
  f = function(x, grp) {
    t_stat = function(x) {
      m = mean(x)
      n = length(x)
      var = sum((x - m) ^ 2) / (n - 1)
      
      list(m = m, n = n, var = var)
    }
    
    g1 = t_stat(x[grp == 1])
    g2 = t_stat(x[grp == 2])
    
    se_total = sqrt(g1$var / g1$n + g2$var / g2$n)
    (g1$m - g2$m) / se_total
  }
  
    apply(X[,-(1:2)], 2, f, X$group)
}

system.time(ttest_hand_calc(X))

   user  system elapsed 
  0.014   0.000   0.015 

Comparison

bench::mark(
  ttest_formula(X, m),
  ttest_for(X, m),
  ttest_apply(X),
  ttest_hand_calc(X),
  check=FALSE
)

# A tibble: 4 × 6
  expression               min   median `itr/sec` mem_alloc `gc/sec`
  <bch:expr>          <bch:tm> <bch:tm>     <dbl> <bch:byt>    <dbl>
1 ttest_formula(X, m) 197.05ms 199.89ms      5.01    8.24MB     26.7
2 ttest_for(X, m)      68.75ms   69.1ms     14.3     1.91MB     26.8
3 ttest_apply(X)       58.24ms  62.82ms     16.1     3.49MB     26.8
4 ttest_hand_calc(X)    8.61ms   9.21ms     89.6     3.45MB     25.9

parallel

Part of the base packages in R

• tools for the forking of R processes (some functions do not work on Windows)

• Core functions:

  • detectCores

  • pvec

  • mclapply

  • mcparallel & mccollect

detectCores

Surprisingly, detects the number of cores of the current system.

detectCores()

[1] 10

pvec

Parallelization of a vectorized function call

system.time(pvec(1:1e7, sqrt, mc.cores = 1))

   user  system elapsed 
  0.016   0.012   0.028 

system.time(pvec(1:1e7, sqrt, mc.cores = 4))

   user  system elapsed 
  0.168   0.151   0.249 

system.time(pvec(1:1e7, sqrt, mc.cores = 8))

   user  system elapsed 
  0.092   0.164   0.152 

system.time(sqrt(1:1e7))

   user  system elapsed 
  0.055   0.018   0.072 

pvec - bench::system_time

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 1))

process    real 
   25ms  24.8ms 

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 4))

process    real 
  150ms   179ms 

bench::system_time(pvec(1:1e7, sqrt, mc.cores = 8))

process    real 
  191ms   223ms 

Cores by size

cores = c(1,4,6,8,10)
order = 6:8
f = function(x,y) {
  system.time(
    pvec(1:(10^y), sqrt, mc.cores = x)
  )[3]
}

res = map(
  cores, 
  function(x) {
     map_dbl(order, f, x = x)
  }
) |> 
  do.call(rbind, args = _)

rownames(res) = paste0(cores," cores")
colnames(res) = paste0("10^",order)

res

          10^6  10^7  10^8
1 cores  0.004 0.032 0.371
4 cores  0.032 0.145 2.006
6 cores  0.024 0.138 1.367
8 cores  0.033 0.127 1.265
10 cores 0.033 0.147 1.548

mclapply

Parallelized version of lapply

system.time(rnorm(1e7))

   user  system elapsed 
  0.262   0.004   0.266 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 2)))

   user  system elapsed 
  0.309   0.095   0.268 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 4)))

   user  system elapsed 
  0.322   0.089   0.161 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 8)))

   user  system elapsed 
  0.336   0.145   0.172 

system.time(unlist(mclapply(1:10, function(x) rnorm(1e6), mc.cores = 10)))

   user  system elapsed 
  0.360   0.150   0.179 

mcparallel

Asynchronously evaluation of an R expression in a separate process

m = mcparallel(rnorm(1e6))
n = mcparallel(rbeta(1e6,1,1))
o = mcparallel(rgamma(1e6,1,1))

str(m)

List of 2
 $ pid: int 14040
 $ fd : int [1:2] 4 7
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"

str(n)

List of 2
 $ pid: int 14041
 $ fd : int [1:2] 5 9
 - attr(*, "class")= chr [1:3] "parallelJob" "childProcess" "process"

mccollect

Checks mcparallel objects for completion

str(mccollect(list(m,n,o)))

List of 3
 $ 14040: num [1:1000000] -0.7172 -1.8334 -0.0983 0.953 0.1629 ...
 $ 14041: num [1:1000000] 0.235 0.554 0.301 0.977 0.644 ...
 $ 14042: num [1:1000000] 0.448 0.192 0.342 1.386 0.397 ...

mccollect - waiting

p = mcparallel(mean(rnorm(1e5)))

mccollect(p, wait = FALSE, 10)

$`14043`
[1] -0.001305267

mccollect(p, wait = FALSE)

NULL

mccollect(p, wait = FALSE)

NULL

doMC & foreach

Packages by Revolution Analytics that provides the foreach function which is a parallelizable for loop (and then some).

• Core functions:

  • registerDoMC

  • foreach, %dopar%, %do%

registerDoMC

Primarily used to set the number of cores used by foreach, by default uses options("cores") or half the number of cores found by detectCores from the parallel package.

options("cores")

$cores
NULL

detectCores()

[1] 10

getDoParWorkers()

[1] 1

registerDoMC(4)
getDoParWorkers()

[1] 4

foreach

A slightly more powerful version of base for loops (think for with an lapply flavor). Combined with %do% or %dopar% for single or multicore execution.

for(i in 1:10) {
  sqrt(i)
}

foreach(i = 1:5) %do% {
  sqrt(i)   
}

[[1]]
[1] 1

[[2]]
[1] 1.414214

[[3]]
[1] 1.732051

[[4]]
[1] 2

[[5]]
[1] 2.236068

foreach - iterators

foreach can iterate across more than one value, but it doesn’t do length coercion

foreach(i = 1:5, j = 1:5) %do% {
  sqrt(i^2+j^2)   
}

[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

[[3]]
[1] 4.242641

[[4]]
[1] 5.656854

[[5]]
[1] 7.071068

foreach(i = 1:5, j = 1:2) %do% {
  sqrt(i^2+j^2)   
}

[[1]]
[1] 1.414214

[[2]]
[1] 2.828427

foreach - combining results

foreach(i = 1:5, .combine='c') %do% {
  sqrt(i)
}

[1] 1.000000 1.414214 1.732051 2.000000 2.236068

foreach(i = 1:5, .combine='cbind') %do% {
  sqrt(i)
}

     result.1 result.2 result.3 result.4 result.5
[1,]        1 1.414214 1.732051        2 2.236068

foreach(i = 1:5, .combine='+') %do% {
  sqrt(i)
}

[1] 8.382332

foreach - parallelization

Swapping out %do% for %dopar% will use the parallel backend.

registerDoMC(4)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))

   user  system elapsed 
  0.298   0.028   0.109 

registerDoMC(8)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))

   user  system elapsed 
  0.302   0.032   0.076 

registerDoMC(10)
system.time(foreach(i = 1:10) %dopar% mean(rnorm(1e6)))

   user  system elapsed 
  0.325   0.042   0.069 

furrr / future

system.time( purrr::map(c(1,1,1), Sys.sleep) )

   user  system elapsed 
  0.000   0.000   3.012 

system.time( furrr::future_map(c(1,1,1), Sys.sleep) )

   user  system elapsed 
  0.053   0.008   3.097 

future::plan(future::multisession) # See also future::multicore
system.time( furrr::future_map(c(1,1,1), Sys.sleep) )

   user  system elapsed 
  0.206   0.007   1.451 

Example - Bootstraping

Bootstrapping is a resampling scheme where the original data is repeatedly reconstructed by taking a samples of size n (with replacement) from the original data, and using that to repeat an analysis procedure of interest. Below is an example of fitting a local regression (loess) to some synthetic data, we will construct a bootstrap prediction interval for this model.

set.seed(3212016)
d = data.frame(x = 1:120) |>
    mutate(y = sin(2*pi*x/120) + runif(length(x),-1,1))

l = loess(y ~ x, data=d)
p = predict(l, se=TRUE)

d = d |> mutate(
  pred_y = p$fit,
  pred_y_se = p$se.fit
)

What to use when?

Optimal use of parallelization / multiple cores is hard, there isn’t one best solution

• Don’t underestimate the overhead cost

• Experimentation is key

• Measure it or it didn’t happen

• Be aware of the trade off between developer time and run time

Statistics and Linear Algebra

An awful lot of statistics is at its core linear algebra.

For example:

• Linear regession models, find

\[ \hat{\beta} = (X^T X)^{-1} X^Ty \]

• Principle component analysis

  • Find \(T = XW\) where \(W\) is a matrix whose columns are the eigenvectors of \(X^TX\).

  • Often solved via SVD - Let \(X = U\Sigma W^T\) then \(T = U\Sigma\).

Numerical Linear Algebra

Not unique to Statistics, these are the type of problems that come up across all areas of numerical computing.

• Numerical linear algebra \(\ne\) mathematical linear algebra

• Efficiency and stability of numerical algorithms matter

  • Designing and implementing these algorithms is hard

• Don’t reinvent the wheel - common core linear algebra tools (well defined API)

BLAS and LAPACK

Low level algorithms for common linear algebra operations

BLAS

• Basic Linear Algebra Subprograms

• Copying, scaling, multiplying vectors and matrices

• Origins go back to 1979, written in Fortran

LAPACK

• Linear Algebra Package

• Higher level functionality building on BLAS.

• Linear solvers, eigenvalues, and matrix decompositions

• Origins go back to 1992, mostly Fortran (expanded on LINPACK, EISPACK)

Modern variants?

Most default BLAS and LAPACK implementations (like R’s defaults) are somewhat dated

• Written in Fortran and designed for a single cpu core

• Certain (potentially non-optimal) hard coded defaults (e.g. block size).

Multithreaded alternatives:

• ATLAS - Automatically Tuned Linear Algebra Software

• OpenBLAS - fork of GotoBLAS from TACC at UTexas

• Intel MKL - Math Kernel Library, part of Intel’s commercial compiler tools

• cuBLAS / Magma - GPU libraries from Nvidia and UTK respectively

• Accelerate / vecLib - Apple’s framework for GPU and multicore computing

OpenBLAS Matrix Multiply Performance

x=matrix(runif(5000^2),ncol=5000)

sizes = c(100,500,1000,2000,3000,4000,5000)
cores = c(1,2,4,8,16)

sapply(
  cores, 
  function(n_cores) 
  {
    flexiblas::flexiblas_set_num_threads(n_cores)
    sapply(
      sizes, 
      function(s) 
      {
        y = x[1:s,1:s]
        system.time(y %*% y)[3]
      }
    )
  }
)